Asymptotic decay of complexity density with depth for random-geometry circuits

Determine whether 1 − C_d (the deficit of complexity density from unity) decays exponentially in circuit depth d for the random-geometry circuits defined by edges of random d-regular graphs with U_ZZ(π/2) two-qubit gates, and characterize the precise asymptotic rate at which C_d approaches 1 as d → ∞.

Background

The paper defines a complexity density C_d,N (normalized effective qubit number) that quantifies the fraction of qubits contributing to the worst-case hardness of exact tensor-network contraction for random circuit sampling. For random-geometry circuits constructed from proper edge-colorings of random d-regular graphs, the authors prove a constant lower bound on C_d as N → ∞ and give an explicit upper bound C_d ≤ 1 − 1/2^d.

While numerics suggest rapid saturation of C_d to near unity at modest depths, the authors highlight that the precise asymptotic approach of C_d to 1 with increasing depth remains unresolved: specifically, whether the deficit 1 − C_d decays exponentially in d.